When I worked in the university genetics lab, we had to do this sort of problem pretty commonly.
Problem: A snake is marketed as a 50% or 66% probability het for a recessive mutant gene. How can you be sure that it is het for that gene?
Answer: Use a breeding test. Mate the possible het to either a known het or to a visual. If any of the babies is a visual, then you are sure the questionable snake actually is a het.
Problem: A snake is marketed as a 50% or 66% probability het for a recessive mutant gene. In a breeding test, it has n babies, none of which is a visual. When do you stop the test?
Answer: In a het x visual mating, half the babies are expected to be visuals, and half the babies are expected to be normal-looking hets. So the probability of any baby being a het is 1/2 (.5). And the probability (P) of there being at least one visual in n eggs is P=1-.5^n. (^n means to the power n.) So the chance of getting 3 normal-looking babies in a clutch of 3 eggs is .5^3 = (.5x.5x.5) = .125, and the chance of getting at least one visual in a clutch of 3 eggs is 1-.125 = .875 or 87.5%. On the other hand, in a normal x visual mating, none of the babies will be visuals. Seven normals and no visuals gives probability of 99% that the possible het is not a het. That 99% point is the place to stop. You will never get a 0% probability that the possible het is a het.
In a possible het x het mating, a quarter (.25) of the babies are expected to be visuals, and the rest (.75) of the babies are expected to be normal-looking. The probability (P) of there being at least one visual in n eggs is P=1-.75^n. Seventeen normals and no visuals gives probability of 99% that the possible het is not a het. That 99% point is the place to stop. You will never get a 0% probability that the possible het is a het.
Breeding a possible het x possible het is a losing proposition unless absolutely necessary. Because the majority of the matings are either het x normal or normal x normal.
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